What is The Area of a Triangle with the Following Vertices GMAT Problem Solving

Question: What is the area of a triangle with the following vertices L(1, 3), M(5, 1), and N(3, 5) ?

1. 3
2. 4
3. 5
4. 6
5. 7

“What is the area of a triangle with the following vertices L(1, 3), M(5, 1), and N(3, 5) ?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “ GMAT Official Guide 2021”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Approach Solution 1: 

It is asked What is the area of a triangle

The Heron's formula, \(\sqrt{(s(s-a)(s-b)(s-c))}\), makes it simple to solve this problem.
S is equal to (a + b + c)/2, where a, b, and c are the distances between any two places, such as LM, MN, and LN. The formula - can be used to get this distance.

\(\sqrt{[x2-x1]^2+[y2-y1]^2}\)

Use the Heron's formula to calculate the triangle's area once you have the values for each side, a, b, and c.

Evaluating further by inserting value we will get
1/2 [(1-15)+(25-3)+(9-5)] = 6

The answer is 6.

Correct Answer: D

Approach Solution 2:

There is another approach to answering this question which is easier
One might use a straightforward formula to determine a triangle's area based on the coordinates of its vertices to address this issue.

Even so, making a diagram will just require a few quick calculations:

Keep in mind that the red triangle's area is 16 minus the areas of the three little triangles in the corners (2*2/2, 4*2/2, and 4*2/2), and the blue square's area is \(4^2\)=16.

Therefore, LMN=16-(2+4+4) = 6 for a triangle's area.

The answer is 6.

Correct Answer: D

Approach Solution 3:

If the vertices o a triangle are :
A (ax, ay), B (bx, by), C (cx, cy)
Then area of ABC is

Area = (ax(by − cy)+ bx (cy − ay)+ cx (ay − by)/ 2
Given that ;
L(1,3), M(5,1), N(3,5)

Area = (1(1−5)+5(5−3)+3(3−1)/ 2

Area = (−4 + (25−15)+(9−3)/ 2

Area = (−4+10+6)/ 2

Area = (−4+16)/ 2

Area = 12/ 2

Area = 6

Correct Answer: D

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